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A342613
a(n) = Sum_{d|n} phi(n/d)^(n+d).
3
1, 2, 17, 34, 4097, 386, 1679617, 263170, 60470273, 20971522, 1000000000001, 67223554, 1283918464548865, 3291294892034, 281543697235969, 2251868534210562, 4722366482869645213697, 4265518198947842, 12748236216396078174437377, 9223671104051085314, 552066177293775007645697, 1100000000000000000000002
OFFSET
1,2
LINKS
FORMULA
a(n) = Sum_{k=1..n} phi(n/gcd(k,n))^(n + gcd(k,n) - 1).
G.f.: Sum_{k>=1} phi(k)^(k+1) * x^k/(1 - phi(k)^(k+1) * x^k).
If p is prime, a(p) = 1 + (p-1)^(p+1).
MATHEMATICA
a[n_] := DivisorSum[n, EulerPhi[n/#]^(n + #) &]; Array[a, 20] (* Amiram Eldar, Mar 17 2021 *)
PROG
(PARI) a(n) = sumdiv(n, d, eulerphi(n/d)^(n+d));
(PARI) a(n) = sum(k=1, n, eulerphi(n/gcd(k, n))^(n+gcd(k, n)-1));
(PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)^(k+1)*x^k/(1-eulerphi(k)^(k+1)*x^k)))
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Mar 16 2021
STATUS
approved